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| | I would like to introduce myself to you, I am Andrew and my spouse doesn't like it at all. I've always cherished living in Mississippi. Credit authorising is how he tends to make money. Doing ballet is some thing she would by no means give up.<br><br>Also visit my blog; best psychics; [http://www.onbizin.co.kr/xe/?document_srl=320614 http://www.onbizin.co.kr/xe/?document_srl=320614], |
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| [[Image:K-space partial Fourier.JPG|thumb|right|300px|The conjugate symmetry of ''k''-space]]'''''k''-space''' is a formalism widely used in [[magnetic resonance imaging]] introduced in 1979 by Likes<ref>{{ cite patent
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| | country = US
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| | number = 4307343
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| | status = patent
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| | title = Moving Gradient Zeugmatography
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| | gdate = 1981-12-22
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| | fdate = 1979-08-20
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| | inventor = Richard S. Likes
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| | assign1 = General Electric Company
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| }}</ref> and in 1983 by Ljunggren<ref>Ljunggren S. Journal of Magnetic Resonance 1983; 54:338.</ref> and Twieg.<ref>{{cite journal | author = Twieg D | title = The k-trajectory formulation of the NMR imaging process with applications in analysis and synthesis of imaging methods. | journal = Medical Physics | volume = 10 | issue = 5 | pages = 610–21 | year = 1983 | pmid = 6646065 | doi = 10.1118/1.595331}}</ref>
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| In [[Physics of magnetic resonance imaging|MRI physics]], ''k''-space is the 2D or 3D [[Fourier transform]] of the MR image measured. Its complex values are sampled during an MR measurement, in a premeditated scheme controlled by a ''pulse sequence'', i.e. an accurately timed sequence of radiofrequency and gradient pulses. In practice, ''k''-space often refers to the ''temporary image space'', usually a matrix, in which data from digitized MR signals are stored during data acquisition. When ''k''-space is full (at the end of the scan) the data are mathematically processed to produce a final image. Thus ''k''-space holds ''raw'' data before ''reconstruction''.
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| ''k''-space is in [[spatial frequency]] domain. Thus if we define <math>k_\mathrm{FE}</math> and <math>k_\mathrm{PE}</math> such that
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| :<math>k_\mathrm{FE}=\bar{\gamma} G_\mathrm{FE}m\Delta t</math> | |
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| :<math>k_\mathrm{PE}=\bar{\gamma} n\Delta G_\mathrm{PE} \tau</math>
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| where FE refers to ''frequency encoding'', PE to ''phase encoding'', <math>\Delta t</math> is the sampling time (the reciprocal of sampling frequency), <math>\tau</math> is the duration of ''G''<sub>PE</sub>, <math>\bar{\gamma}</math> (''gamma bar'') is the [[gyromagnetic ratio]], ''m'' is the sample number in the FE direction and ''n'' is the sample number in the PE direction (also known as ''partition number''), the 2D-[[Fourier Transform]] of this encoded signal results in a representation of the spin density distribution in two dimensions. Thus position (''x'',''y'') and spatial frequency (<math>k_\mathrm{FE}</math>, <math>k_\mathrm{PE}</math>) constitute a Fourier transform pair.
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| Typically, ''k''-space has the same number of rows and columns as the final image and is filled with raw data during the scan, usually one line per TR (Repetition Time).
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| An MR image is a complex-valued map of the spatial distribution of the transverse magnetization ''M''<sub>xy</sub> in the sample at a specific time point after an excitation. Conventional qualitative interpretation of [[Fourier Analysis]] asserts that low spatial frequencies (near the center of ''k''-space) contain the [[signal to noise]] and [[contrast (vision)|contrast]] information of the image, whereas high spatial frequencies (outer peripheral regions of ''k''-space) contain the information determining the [[image resolution]]. This is the basis for advanced scanning techniques, such as the ''keyhole'' acquisition, in which a first complete ''k''-space is acquired, and subsequent scans are performed for acquiring just the central part of the ''k''-space; in this way, different contrast images can be acquired without the need of running full scans.
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| A nice symmetry property exists in ''k''-space if the image magnetization ''M''<sub>xy</sub> is prepared to be proportional simply to a contrast-weighted proton density and thus is a real quantity. In such a case, the signal at two opposite locations in ''k''-space is:
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| :<math>S(-k_\mathrm{FE},-k_\mathrm{PE}) = S^*(k_\mathrm{FE},k_\mathrm{PE}) \,</math> | |
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| where the star (<math>^*</math>) denotes [[complex conjugation]].
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| Thus ''k''-space information is somewhat redundant then, and an image can be reconstructed using only one half of the ''k''-space, either in the PE (Phase Encode) direction saving scan time (such a technique is known as ''half Fourier'' or ''half scan'') or in the FE (Frequency Encode) direction, allowing for lower sampling frequencies and/or shorter echo times (such a technique is known as ''half echo''). However, these techniques are approximate due to phase errors in the MRI data which can rarely be completely controlled (due to imperfect [[Shim (magnetism)|static field shim]], effects of spatially selective excitation, signal detection coil properties, motion etc.) or nonzero phase due to just physical reasons (such as the different chemical shift of fat and water in gradient echo techniques).
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| ==References==
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| <references/>
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| ===Further reading=== | |
| *McRobbie D., et al. ''MRI, From picture to proton.'' 2003
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| *Hashemi Ray, et al. ''MRI, The Basics'' 2ED. 2004.
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| [[Category:Magnetic resonance imaging]]
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I would like to introduce myself to you, I am Andrew and my spouse doesn't like it at all. I've always cherished living in Mississippi. Credit authorising is how he tends to make money. Doing ballet is some thing she would by no means give up.
Also visit my blog; best psychics; http://www.onbizin.co.kr/xe/?document_srl=320614,